closure of a vector subspace in a normed space is a vector subspace

First of all, 0∈S¯ because 0∈S. Now, let x,y∈S¯, and λ∈K (where K is the ground field of the vector spaceV). Then there are two sequences in S, say (xn)n∈ℕ and (yn)n∈ℕ which converge to x and y respectively.

Then, the sequence (xn+λ⋅yn)n∈ℕ is a sequence in S (because S is a vector subspace), and it’s trivial (use properties of the norm) that this sequence converges to x+λ⋅y, and so this sum is a vector which lies in S¯.